This invention relates to the construction of selective frequency filters, specifically to configurations and methodologies suitable for their implementation using monolithic capacitors and monolithic planar inductors with low Q (factor of merit) such as those available in standard integrated circuits technologies.
The mathematical behavior of frequency filters is described with a transfer function that depends on the complex frequency (s) and a number of complex constants. This transfer function has the form:
                              H          ⁡                      (            s            )                          =                  (                                                    (                                  s                  -                                      z                    0                                                  )                            ⁢                              (                                  s                  -                                      z                    1                                                  )                            ⁢                                                          ⁢              …              ⁢                                                          ⁢                              (                                  s                  -                                      z                    N                                                  )                                                                    (                                  s                  -                                      p                    0                                                  )                            ⁢                              (                                  s                  -                                      p                    1                                                  )                            ⁢                                                          ⁢              …              ⁢                                                          ⁢                              (                                  s                  -                                      p                    M                                                  )                                              )                                    (        1        )            where N≦M for any passive network. The complex constants zi are called the zeros and the pj are the poles. As can be seen from (1), both the values of the poles and zeros determine the shape of the transfer function. If the network is passive and loss-less, the zeros are imaginary numbers (the real part equals zero). The poles are complex numbers where the real part is negative. The ratio between the imaginary and real parts of the poles is called the Q (to be distinguished from the Q of the electrical components implementing the filter).
One of the most common passive filter architectures is the doubly terminated loss-less LC ladder. This topology is based on a network of reactive components, such as inductors and capacitors, inserted between termination resistors. A multiplicity of signal generators are connected at the source side, and the resulting filtered signal is obtained at the load side.
The functionality of these filters is based on the fact that across the passband region there is maximum power transfer through the reactive network, and in the stopband, most of the signal energy is reflected back to the source end.
For loss-less LC ladders, there is an elegant and powerful synthesis tool for the design. This mathematical procedure yields the values of the components as a function of the poles and zeros of the transfer function. Synthesis procedures have been extensively documented, such as for example in the book by A. S. Sedra and P. O. Brackett: Filter Theory And Design: Active and Passive. Also, there are several computer programs available implementing synthesis, FILTOR2 being one of them. On the other hand, when the components are lossy, as shown below, the accepted solution is to introduce active circuitry in order to compensate for the losses. In this fashion, the combined arrangement of the lossy component and the active cancellation circuitry yields the equivalent of loss-less components, as described in the article by D. L. Li and Y. Tsividis: “Design Techniques for Automatically Tuned Integrated Gigahertz-Range Active Filters”, IEEE J. Solid-State Circuits, vol. SC-37, pp. 967-977, August 2002.
For a real inductor, the Q at a given angular frequency ω is given by:
                              Q          0                =                  (                                                    ω                0                            ⁢              L                        r                    )                                    (        2        )            where L is the inductance and r is the series resistance. For the case of a capacitor, the Q is:
                              Q          0                =                  (                                                    ω                0                            ⁢              C                        g                    )                                    (        3        )            where C is the capacitance and g is the parallel conductance. In a loss-less component, the Q0 in (2) and (3) is infinity.
The most common type of capacitors used in silicon integrated circuits is the planar type. These can have fixed or variable values of capacitance. The fixed type is built by stacking two or more layers or plates of a conductive material such as metal, separated by one or more layers of insulating material known as dielectric. The most common variable type, also referred to as varactor, changes its value as a function of a bias voltage applied across the plates. This type can be built with a semiconductor junction, or using the nonlinear properties of metal-oxide-silicon (MOS) interfaces. Variable capacitors are used to vary or adjust the spectral characteristics of filters by means of one or more bias voltages applied across one or more capacitors in the network. The Q's of integrated planar capacitors can reach values in the 100's.
A planar inductor in silicon integrated technology is built as a spiral of a conductive material, such as aluminum or copper, set on top of insulating material, generally silicon dioxide (Si2O). The shape of the spiral can be made circular or polygonal. Two or more inductors can be placed in physical proximity forming a set of coupled-inductors (a transformer). The Q of these planar inductors rarely achieves values greater than 20. This Q is too low to be considered a low-loss component suitable for the implementation with conventional synthesis methods. Thus inductors are the main roadblock for the construction of highly selective filters using conventional synthesis methods.
M. Dishal in his paper: “Design of dissipative band-pass filters producing desired Transformers for Si RF IC's”, Proc. Inst. Radio Eng., vol. 37, pp. 1050-1069; September 1949 shows a methodology for the design of frequency filters using lossy components without any active cancellation. It is based on mathematically compensating for the finite Q of the resonators in the calculation of the filter component values. However, his filter configurations are still based on-loss-less ladders that include the termination resistors. There is a quantitative ingredient in his line of work that pertains to the fact that the Q of his resonators are much larger than any Q in the desired transfer function. In short, Dishal's work demonstrates a way to compensate for moderate to low losses of the components in a conventional filter architecture.
Given the convenience of existing computer synthesis tools, designers are currently using circuitry such as those described by Li et al, aimed at canceling the losses to achieve nearly loss-less inductors by nulling the denominator of (2), as illustrated in FIGS. 1 and 1A. FIG. 1 shows a resonator 1 with a lossy inductor of value L, where the loss is represented by the series resistor r as indicated in (2). The obvious cancellation configuration would be to place a negative resistor in series to null the total resistance. However, because of some limitations in active circuits, it is considerably simpler and more feasible to use a parallel negative resistor Rc to achieve cancellation. This equivalent negative resistance can be implemented with the active circuit 2 shown in FIG. 1A. The circuit formed by M1 and M2 is a simplified example of a cross-coupled pair in which the magnitude of negative resistance can be set by controlling the tail current of the cross-coupled pair through the control voltage VCTL. At a given angular frequency ω, the value of this parallel cancellation resistor has to be equal to:
                              R          C                =                  -                      r            ⁡                          (                              1                +                                                                            ω                      2                                        ⁢                                          L                      2                                                                            r                    2                                                              )                                                          (        4        )            
The main problems associated with active, loss-cancellation circuitry are the following:                (1) the cancellation circuit can use a significant amount of silicon area (die area) and consume considerable power;        (2) the unavoidable non-linearity of the cancellation circuitry creates a response that is a function of the signal amplitude, thus making many of the most critical specifications of related applications very hard to meet;        (3) the transistors in the cancellation circuitry introduce noise on the signal path;        (4) an additional complex control architecture is needed for adjusting the amount of cancellation, and inaccuracies of this cancellation could yield unstable operating conditions;        (5) this control circuitry further consumes large amounts of power and die area; and        (6) as shown in (4), because of the dependency on ω, the cancellation achieved does not extend evenly through the whole passband region, thus creating inaccuracies in the resulting response.        